The Art and Craft of Mathematical Problem Solving

152 Pages·2012·0.8 MB·English

by  Paul Zeitz

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Professor Paul Zeitz
University of San Francisco
Professor Paul Zeitz is a specialist in mathematical
problem solving. A Professor of Mathematics at the
University of San Francisco, he won the USA Mathematical
Olympiad and was a member of the first American team
to participate in the International Mathematical Olympiad.
Since then, he has helped train several American
International Mathematical Olympiad teams, most notably
the 1994 team—the first to achieve a perfect score.
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Course No. 1483 © 2010 The Teaching Company. PB1483A k
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The Teaching Company.
Paul Zeitz, Ph.D.
Professor of Mathematics
University of San Francisco
P
aul Zeitz is Professor of Mathematics at the
University of San Francisco. He majored in
History at Harvard University and received
a Ph.D. in Mathematics from the University of
California, Berkeley, in 1992, specializing in
Ergodic Theory. Between college and graduate school, he taught high school
mathematics in San Francisco and Colorado Springs.
One of Professor Zeitz’s greatest interests is mathematical problem
solving. He won the USA Mathematical Olympiad (USAMO) and was
a member of the (cid:191) rst American team to participate in the International
Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and
edited problems for several national math contests, including the USAMO.
He has helped train several American IMO teams, most notably the 1994
“Dream Team,” which was the (cid:191) rst—and heretofore only—team in history
to achieve a perfect score. This work, and his experiences teaching at USF,
led him to write The Art and Craft of Problem Solving (Wiley, 1999;
2nd ed., 2007).
Professor Zeitz has also been active in events for high school students. He
founded the San Francisco Bay Area Math Meet in 1994; cofounded the Bay
Area Mathematical Olympiad in 1999; and currently is the director of the
San Francisco Math Circle, a program that targets middle and high school
students from underrepresented populations.
Professor Zeitz was honored in March 2002 with the Award for Distinguished
College or University Teaching of Mathematics from the Northern California
Section of the Mathematical Association of America (MAA), and in January
2003, he received the MAA’s national teaching award, the Deborah and
Franklin Tepper Haimo Award. (cid:374)
i
Table of Contents
INTRODUCTION
Professor Biography ............................................................................i
Course Scope .....................................................................................1
LECTURE GUIDES
LECTURE 1
Problems versus Exercises ................................................................4
LECTURE 2
Strategies and Tactics ......................................................................10
LECTURE 3
The Problem Solver’s Mind-Set ........................................................16
LECTURE 4
Searching for Patterns ......................................................................20
LECTURE 5
Closing the Deal—Proofs and Tools .................................................24
LECTURE 6
Pictures, Recasting, and Points of View ...........................................30
LECTURE 7
The Great Simpli(cid:191) er—Parity .............................................................35
LECTURE 8
The Great Uni(cid:191) er—Symmetry ..........................................................41
LECTURE 9
Symmetry Wins Games! ...................................................................45
LECTURE 10
Contemplate Extreme Values ...........................................................51
ii
Table of Contents
LECTURE 11
The Culture of Problem Solving........................................................55
LECTURE 12
Recasting Integers Geometrically.....................................................58
LECTURE 13
Recasting Integers with Counting and Series...................................63
LECTURE 14
Things in Categories—The Pigeonhole Tactic ..................................67
LECTURE 15
The Greatest Uni(cid:191) er of All—Invariants .............................................71
LECTURE 16
Squarer Is Better—Optimizing 3s and 2s .........................................76
LECTURE 17
Using Physical Intuition—and Imagination .......................................80
LECTURE 18
Geometry and the Transformation Tactic ..........................................85
LECTURE 19
Building from Simple to Complex with Induction ..............................89
LECTURE 20
Induction on a Grand Scale ..............................................................94
LECTURE 21
Recasting Numbers as Polynomials—Weird Dice ............................98
LECTURE 22
A Relentless Tactic Solves a Very Hard Problem ...........................103
LECTURE 23
Genius and Conway’s In(cid:191) nite Checkers Problem ..........................108
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Table of Contents
LECTURE 24
How versus Why—The Final Frontier.............................................113
SUPPLEMENTAL MATERIAL
Solutions .........................................................................................117
Timeline ..........................................................................................128
Glossary .........................................................................................131
Biographical Notes .........................................................................137
Bibliography ....................................................................................139
iv
The Art and Craft of Mathematical Problem Solving
Scope:
T
his is a course about mathematical problem solving. The phrase
“problem solving” has become quite popular lately, so before we
proceed, it is important that you understand how I de(cid:191) ne this term.
I contrast problems with exercises. The latter are mathematical questions that
one knows how to answer immediately: for example, “What is 3 + 8?” or
“What is 3874?” Both of these are simple arithmetic exercises, although the
second one is rather dif(cid:191) cult, and the chance of getting the correct answer is
nil. Nevertheless, there is no question about how to proceed.
In contrast, a problem is a question that one does not know, at the outset, how
to approach. This is what makes mathematical problem solving so important,
and not just for mathematicians. Arguably, all pure mathematical research
is just problem solving, at a rather high level. But the problem-solving
mind-set is important for all who take learning seriously, especially lifelong
learners. Much of the current craze in brain strengthening focuses merely on
exercises. These are not without merit—indeed, mental exercise is essential
for everyone—but they miss out on a crucial dimension of intellectual life.
Our brains are not just for doing crosswords or sudoku—they also can and
should help us with intensive contemplation, open-ended experimentation,
long wild goose chases, and moments of hard-earned triumph. That is what
problem solving is all about.
An analogy that I frequently use compares an exerciser to a gym rat and a
problem solver to a mountaineer. The latter’s experience is riskier, messier,
dirtier, less constrained, less certain, but much more fun. For those of you
who prefer more civilized pursuits, consider 2 ways to learn Italian. One
involves toiling over grammar exercises and translations of texts. The other
method is to spend a few months, perhaps after a short bit of preparation,
in a small town in Italy where no one else speaks English. Again, the latter
approach is messier but fundamentally richer.
1
Becoming a good problem solver requires new skills (mathematical as well
as psychological) and patient effort. My pedagogical philosophy is both
experiential and analytic. In other words, you cannot learn problem solving
without working hard at lots of problems. But I also want you to understand
what you are doing at as high a level as possible. We will break down the
process of solving a problem into investigation, strategy, tactics, and (cid:191) ner-
grained tools, and we will often step back to discuss not just how we solved
a problem but why our methods worked.
Problems, by de(cid:191) nition, are hard to solve. Solving problems requires
investigation, and successful investigations need strategies and tactics.
Strategies are broad ideas, often not just mathematical, that facilitate
investigation. Some strategies are psychological, others organizational, and
others simply commonsense ideas that apply to problems in any (cid:191) eld. Tactics
are more narrowly focused, mostly mathematical ideas that help solve many
problems that have been softened by good strategy. Additionally, there are
very specialized techniques, called tricks by some, that I call tools.
This course is devoted to the systematic development of investigation
methods, strategies, and tactics. Besides this “problemsolvingology,” I
will introduce you to mathematical folklore: classic problems as well as
mathematical disciplines that play an important role in the problem-solving
world. For example, no course on problem solving is complete without some
discussion of graph theory, which is an important branch of math on its own
but is also a very accessible laboratory for exploring problem-solving themes.
Many of the lectures will include small amounts of new mathematics that
we will build up and stitch together as the course progresses. The topics are
largely drawn from discrete mathematics (graph theory, integer sequences,
number theory, and combinatorics), because this branch of math does not
require advanced skills such as calculus. That does not mean it is easy, but
we will move slowly and develop new ideas carefully.
A small but important part of the course explores the culture of problem
solving. I will draw on my experience as a competitor, coach, and problem
writer for various regional, national, and international math contests, to make
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p the little-known world of math Olympiads come to life. And I will discuss
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the recent educational reform movement (in which I am a key player) to
bring Eastern European–inspired mathematical circles to the United States.
Problem solving is not a vertically organized discipline; it is not something
that one learns in a linear fashion. Thus the overall organization of this
course has a recursive, spiral nature. The (cid:191) rst few lectures introduce the main
ideas of strategy and tactics, which then are revisited and illuminated by
different examples. We will often return to and re(cid:191) ne previously introduced
ideas. Overall, the topics get more complex toward the end of the course,
but the underlying concepts do not really change. An analogy is a theme
and variations musical piece, where the main theme is introduced with a
slow, stately rhythm and later ends in complex avant-garde interpretations.
By the end of the course, you should understand the main theme (the basic
and powerful strategies and tactics of problem solving) quite well because
you had to struggle with the complex interpretations (the advanced folklore
problems that used the basic strategies in novel ways).
Problem solving is not just solving math problems. It is a mental discipline;
successful investigations demand concentration and patient contemplation
that few of us can do, at least at (cid:191) rst. Also, problem solving is an aesthetic
discipline—in other words, an art—where we create and contemplate objects
of elegance and beauty. I hope that you enjoy learning about this wonderful
subject as much as I have! (cid:374)
3
Problems versus Exercises
Lecture 1
There is always a porous boundary between problem and exercise, but
a problem by its very nature requires investigation, sometimes very
intense and sustained investigation. The investigation of a problem
employs strategies and tactics, and that’s what this course is about.
I
n this introductory lecture, we de(cid:191) ne the main entity that we will study
in this course: problems. Problems, by de(cid:191) nition, are dif(cid:191) cult, and our
investigation of them cannot proceed without organized strategies and
tactics. Indeed, our course focuses on 3 things: investigation, strategies, and
tactics. Problem solving is at the heart of mathematics. It is not just a way of
thinking about math but is an intellectual lifestyle with its own mathematical
folklore and culture. Most of our learning
will be by example. Almost every lecture
will revolve around one or more problems. Much of the current
You, the viewer, will need to use the Pause craze in brain
button and pencil and paper. This lecture strengthening focuses
will include several fun problems not
on exercises. This
requiring any special mathematical skills,
is not without merit,
but in later lectures, the problems will be
s more complex. but our brains should
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s also work on intensive
erci Who am I, and what do I do? I have contemplation
x been a professor at the University of San
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bl I (cid:191) rst learned about problem-solving mathematics as a mathlete at Stuyvesant
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1: international competitions, some of the most formative experiences of my
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ct school teachers, run math clubs, write problems for math competitions, and
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